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LECTURE 2:ELECTRON EMISSION AND CATHODE EMITTANCE

DAVID H. DOWELLSTANFORD LINEAR ACCELERATOR CENTER

Abstract. The objectives of this lecture are to define the basic electron emis-sion statistics, describe the electrical potentials at the cathode surface, define

the thermal emittance and derive the cathode emittance for thermal, photo-

electric and field emission.

1. Introduction

The electron density inside a cathode is many orders of magnitude higher thanthat of the emitted electron beam. This is seen by considering that the densityof conduction band electrons for metals is 1022 to 1023 electrons/cm3, roughly oneelectron per atom. Whereas the density of electrons in a 6 ps long, 200 microndiameter cylindrical bunch with 1 nC of charge is 1.1×1014 electrons/cm3. Thusthe transition from bound to free reduces the electron density by eight to nine ordersof magnitude. In addition, the energy spread, or thermal energy of the electronsinside the cathode material is low. For example, in copper the energy spread nearthe Fermi energy is kBT or 0.02 eV at room temperature (300 degK). However,in order to release these cold, bound electrons, one needs to heat the cathode toapproximately 2500 degK, resulting in a beam with a thermal energy of 0.20 eV.

In general, the electron temperature of the emission process determines the fun-damental lower limit of the beam emittance. This ultimate emittance is oftencalled the thermal emittance, due to the Maxwell-Boltzmann (MB) distribution ofthermionic emitters. Strictly speaking, the term ’thermal emittance’ applies only tothermionic emission, but the concept of thermal emittance or the intrinsic cathodeemittance can be applied to all three fundamental emission processes:

1. thermionic emission,2. photo-electric emission and3. field emission.This lecture begins with definitions of Maxwell-Boltzmann and Fermi-Dirac sta-

tistics, and discusses the electric fields at the cathode surface which the electronneeds to overcome to escape. Then the physics of each of the three emission pro-cesses is described and their cathode emittances are derived.

2. Electron Statistics and the Emission Processes

Elementary particles in general can be classified as either bosons or fermions de-pending upon whether they have integer or half integer spin, respectively. Bosonsobey classical Maxwell-Boltzmann statistics, while fermions follow Dirac-Fermi sta-tistics. These statistics define the probability a particle occupies an given energy

1

2 DAVID H. DOWELL STANFORD LINEAR ACCELERATOR CENTER

state based on the the distribution of N-particles on k-energy intervals for the twoparticle types:

1. particles any number of which can share the same energy state, follow theMaxwell-Boltzmann distribution.

2. particles which cannot share the same energy state having only one particleper energy state, follow Fermi-Dirac distribution.

The first particle type obeys classical Maxwell-Boltzmann (M-B) statistics withthe energy distribution of occupied states given by,

(1) fMB = e−E/kBT

For the second particle type, of which electrons are are a member being fermions,the energy distribution of occupied states (DOS) is given by the Fermi-Dirac (F-D)function,

(2) fFD =1

1 + e(E−EF )/kBT

The distributions are compared in Figure 1, showing they have very nearly thesame high energy tails. During thermionic emission the cathode is heated to hightemperature to increase the high energy tail of the distribution and promote emis-sion. In this case, M-B statistics is completely valid and the classical concept oftemperature applies. However, as will be shown, photo-electric and field emissioninvolves the excitation of electrons from below the Fermi energy and not only thetail of the distribution. In these cases, F-D statistics should be used.

Figure 1. Comparison of the particle energy distributions in thehigh-energy tails of the classical Maxwell-Boltzmann and the quan-tum mechanical Fermi-Dirac functions.

In some applications electrons are produced using two of more of these threeprocesses. A good example is field emission, which is usually considered to occurat low temperature, but in practice is often combined with thermionic emission inthe high field of thermionic RF guns. And there are proposals for using all threeprocesses as in photo-assisted thermionic cathodes for high-field RF guns.

3. Fields Near the Cathode Surface

An important aspect of electron emission concerns the fields and potentials atand extremely near the cathode surface, and how these fields affect the electron

LECTURE 2: ELECTRON EMISSION AND CATHODE EMITTANCE 3

states. The electric potential energy as a function of distance from the cathode isgiven by

(3) eΦ = eφwork −e2

16πε0x− eE0x

which is the sum of the work function, φwork, the image charge potential and theapplied electric field, E0.

Comparison of the electric fields and the energy density of occupied states neara metal-vacuum boundary is shown in Figure 2. The total external potential, Φ, isthe sum of applied and image fields for a single electron, and peaks approximately2 nm from the surface. Electrons can escape with energies greater than the workfunction or those with lower energy can tunnel through the barrier. In the cases ofthermionic and photo-emission, the escaping electrons must have energies greaterthan the barrier. In field emission, electrons tunnel through the barrier. This is animportant distinction: In the first two the electrons are excited above the barrierto escape, while in field emission the barrier height is lowered by an external fieldto encourage tunneling.

Figure 2. The potential energy barrier near the cathode surface(right) and the density of occupied states (left) are plotted within10 nm of a metallic surface. The work function barrier is producedby an excess of electrons spilling out from inside the cathode toa distance of less than an angstrom. The image potential is for asingle electron and the applied field corresponds to 100 V/micron.

The temperature of the electron gas in the bulk material affects the probabilityof emission and the emitted electron energy distributions for all three phenomena.The reduction of the barrier by the applied field is called the Schottky effect andplays a central role in all emission processes, especially field emission. Photo-electricemission promotes electrons directly from the energy region near the Fermi energy.In the following sections, the underlying physics for these three processes will bedescribed.


4. Thermionic Emission

In order for an electron to escape a metal it needs to have sufficient kinetic inthe direction of the barrier to overcome the work function,

(4)mv2

x

2> eφwork ⇒ vx >

√2eφworkm

Assume that the cathode has an applied electric field large enough to remove allelectrons from the surface, so there is no space charge limit for emission, but stilllow enough as to not affect the barrier height. Then the thermionic current densityfor a cathode at temperature T is given by,

(5) jthermionic = n0e〈vx〉 = n0e

∫vx>

√2eφwork

m

vxfFDd~v

The region of integration over the F-D distribution is shown in Figure 3.As discussed above, the interactions involving the high energy electrons in the tail

of the Fermi-Dirac density of states (The F-D distribution is essentially identical tothe classical M-B for energies greater than 1.005EF , as shown in Figure 1.), allowsit’s replacement with the classical, Maxwell-Boltzmann distribution,

(6) fMB = e−E/kBT

The equation for the current density is then,(7)

jthermionic = n0e

∫vx>

√2eφwork

m

vxfMBd~v = n0e

∫vx>√

2eφwork/m

vxe−m(v2x+v2y+v2z)

2kBT d~v

Performing these simple integrals gives the thermionic current density,

(8) jthermionic = 2n0e

(2kBTm

)2

e−φwork/kBT

Or regrouping the leading constants, gives the Richardson-Dushman (R-D) equationfor thermionic emission [Reiser, p 8],

(9) jthermionic = A(1− r)T 2e−φwork/kBT

Here A is 120 amp/cm2/degK2, and (1-r) accounts for the reflection of electronsat the metal surface. The reflection and refraction of electrons as they transit thesurface is discussed in a later section. In terms of fundamental quantities, theuniversal constant A is [”Solid State Physics”, by Ashcroft and Mermin, p. 363]

(10) A = − em

2π2~3

The exponential dependence upon temperature of the R-D equation illustrateshow thermionic current rises rapidly with temperature, and with decreasing workfunction. This same work function also applies to photo-electric and field emission.

The velocity distribution for thermally emitted electrons is obtained from thederivative of Maxwell-Boltzmann particle distribution,

(11)1ne

dn(vx)dvx

=m

kBTvxe

−mv2x2kBT

Figure 4 shows the energy distribution assuming the emitted electrons obey Maxwell-Boltzmann statistics. At room temperature, the mean transverse velocity is 9×104


Figure 3. Fermi-Dirac energy distributions for thermionic emis-sion. Electrons in the high energy tail of the distribution (energiesgreater than the work function) are thermally emitted for cathodetemperatures of 2500 (red-pink) and 3000 (blue-aqua) degK.

m/s, or 0.025 eV. The initial spread in transverse velocity due to the electron tem-perature gives the beam angular divergence and hence its thermionic emittance.

Following Lawson[Lawson,p. 209], we assume the normalized emittance is evalu-ated close to the cathode surface where the electron flow is still laminar (no crossingof trajectories) and any correlation between position and angle can be ignored. Inthis case, normalized cathode emittance is given by,

(12) εN = βγσxσx′

The root-mean-square (rms) beam size, σx, is given by the transverse beamdistribution which for a uniform radial distribution with radius Rc is Rc/2. Therms divergence is given by

(13) σx′ =〈px〉ptotal

=1βγ

√〈v2x〉

c

The normalized, rms thermal emittance is then

(14) εN = σx

√〈v2x〉

cThe mean squared transverse velocity for a Maxwell velocity distribution is,

(15) 〈v2x〉 =

∫∞0v2xe− mv2x

2kBT dvx∫∞0e− mv2x

2kBT dvx

=kBT

m

Therefore the thermionic emittance of a Maxwell-Boltzmann distribution at tem-perature, T , is

(16) εthermionic = σx

√kBT

mc2


Figure 4. Maxwell-Boltzmann electron energy distributions at300 degK where the rms electron energy spread is 0.049 eV, andat 2500 degK corresponding to an rms energy spread of 0.41 eV.

The divergence part of the cathode emittance contains all the physics of boththe emission process and the cathode material properties and as such summarizesmuch of the interesting physics of the emission process. The beam size in coordinatespace simply traces out the angular distribution to form the transverse phase spacedistribution as illustrated in Figure 5. Given that σx depends upon the particulartransverse distribution being used, there is often a serious ambiguity which ariseswhen expressing the thermal emittance in terms of ”microns/mm”. The confusionresults in not knowing whether rms or flat top radii are used for the transverseradius. Therefore we suggest quoting a quantity called the normalized divergence,which for thermionic emission is

(17) ∆thermionic ≡√kBT

mc2

The thermal normalized divergence and the rms energy spread for the M-B distri-bution as a function of temperature is given in Figure 6.

Class exercise: Compute the thermionic emittance for a distribution of hotspots.

5. Photo-electric emission

Photoelectric emission from a metal can be described by the three steps of theSpicer model [Spicer]:

1. Photon absorption by the electron


Figure 5. Each point on the cathode surface emits electrons ac-cording to the M-B velocity distribution, mapping into a ’line’ ofphase space points along the divergence axis. The emittance isthen linear in the beam size with a divergence given by the M-Bdistribution at temperature T.

Figure 6. The normalized divergence and rms energy spread ofthe M-B gas as a function of temperature.

2. Electron transport to the surface3. Escape through the barrierThe kinematics of the first step is illustrated in Figure 7 showing the location of

the vacuum state at the work function minus the Schottky energy. This Schottkyenergy accounts for the lower barrier when a large external field is applied to thecathode, and is typically a few ten’s of an eV for fields of 100 MV/m. This schottkyeffect can significantly increase the quantum efficiency since the work function fora metal cathode like copper is 4.6 eV.

Some insight into the photo-emission process can be gained by assuming thatall the electrons absorbing photons in step 1 escape. Then the quantum efficiencyis simply proportional to the number of electrons the photon can excite above thebarrier as indicated by the shaded region in the Figure 8. In addition the energyspread and hence the photo-electric emittance is also proportional to the photon


Figure 7. Spicer’s three-step model of photoemission.

energy. Therefore it appears that the higher the QE, the larger the photo-electricemittances. This is indeed true, as will be demonstrated below.

Figure 8. Kinematics of photo-emission.

During step 2, transport to the surface, the electron-electron scattering redis-tributes the energy distribution on a fast time scale, while the slower electron-phonon interaction acts on the electron through the lattice atoms. In general,the electron-phonon interaction is important for semi-conductor cathodes whileelectron-electron scattering is dominant for metals. These scattering interactions


make step 2 a complex process with a time-dependence Fermi-Dirac temperatureand a direct high-energy component. In one experiment, the time for the electronsto thermalize among themselves was measured in a laser pump-probe measurement[Fann et al., ”Observation of the thermalization of electrons in a metal excited byfemtosecond optical pulses,” in Ultrafast Phenomenona, ed. J.-L. Martin, A. Mei-gus, G.A. Mourou and A.H. Zewail, Springer Verlag 1993, p331-334.]. Here thephotoemission energy spectra were measured as a function of time delay betweenan IR pump laser and a UV probe. The electrons absorbed energy from the IRpulse, and having too low an energy to escape could only scatter and thermalizewith other electrons and phonons. The UV probe sampled the electron energydistribution as a function of time by photo-emitting the electrons excited by theIR pulse. Fann’s results are reproduced in Figure 9 and illustrate how the Step1, high-energy electrons thermalize their energy on a time scale of approximately400 fs, achieving a maximum Fermi-Dirac temperature of 700 degK. This experi-ment not only determined the electron-electron relaxation time, but also gives thethermodynamic parameters for the electronic equation-of-state.

Figure 9. Electron photo-emission spectra on the sub-ps timescale for Au.

Not only does the electron need to have enough energy to escape, but is mustalso be traveling toward the cathode surface. Therefore, the electron’s momentumperpendicular to the surface, pz, needs to satisfy,

(18)p2z

2m> EFermi + φwork − φSchottky

The longitudinal momentum is pz = ptotcosθ =√

2m(E + ~ω)cosθ, therefore themaximum angle of emission, max, satisfying the above condition is

(19) cosθmax =

√EF + φwork − φSchottky

E + ~ωwhere EF is the cathode Fermi energy, φwork is the work function for electronemission, φSchottky is the reduction in the barrier potential due to the external


field, E is the electron’s initial energy inside the cathode and is the photon energy.The relations between energy, momenta and angles at the surface are given inFigure 10. It’s relevant to note that these momenta are for electrons just inside thecathode. The momenta outside the metal are discussed in a later section.

Figure 10. Definition of the maximum angle of emission for elec-trons at the cathode-vacuum boundary.

While the electron moves toward the surface, it can scatter with other electronsor with the lattice and excite phonons, lose energy and not escape the cathode. In ametal cathode, electron-electron scattering has a larger cross section than electron-phonon scattering. The mean free path for fermi gas of electrons can be computedby realizing that the Pauli principle leads to blocking of final states the electronscatter to and an relatively slow energy dependence of (E−EF )−3/2 near the Fermienergy [W. F. Krolikowski and W. E. Spicer, Phys. Rev. 185, 1969, 882-900]. Themean free path for electrons excited 4 to 5 eV above the Fermi energy is 45 to 70angstroms. Since the absorption depth for a UV photon in copper is approximately120 angstroms, the electrons absorbing photons at the greatest depth can on averagescatter two to three times before reaching the surface. After one scattering, theelectron loses too much energy to escape.

Putting all these elements together, the following expression for the QE is found[D. H. Dowell, K.K. King, R.E Kirby and J.F. Schmerge, ”In situ cleaning ofmetal cathodes using a hydrogen beam,” PRST-AB 9, 063502 (2006)] where φeff =φwork − φSchottky,

(20) QE(ω) = (1−R(ω))

∫ EFEF+φeff−~ω F (E)dE

∫ 1√EF+φeffE+~ω

d(cosθ)∫ 2π

0dϕ∫

dE∫d(cosθ)

∫dϕ

After some mathematics one finds,

(21) QE(ω) =(1−R(ω))

1 + λoptλe−e

EF + ~ω2~ω

(1−

√EF + φeffEF + ~ω

)2

Where R(ω) is the reflectivity, λopt is the photon’s optical depth and λe−e is theelectron mean free path.


The normalized divergence can be written in terms of the mean square of thetransverse momentum which in turn is related to the electron distribution function,g(E, θ, ϕ), just inside the cathode surface,

(22) 〈p2x〉 =

∫ ∫ ∫g(E, θ, ϕ)p2

xdEd(cosθ)dϕ∫ ∫ ∫g(E, θ, ϕ)dEd(cosθ)dϕ

The g-function and the integration limits depend upon the emission processes. Forthe three-step photo-emission model, g depends only on energy,

(23) gphoto = (1− fFD(E + ~ω))fFD(E)

Which is the mathematical expression for the electron distribution just below theFermi energy shown as shaded in Figure 8. Since the Fermi-Dirac function, fFD, attemperatures near ambient is well-represented by a Heaviside-step function whichdetermines the limits on the energy integration. The θ-integration ranges fromzero to θmax, as given by Equation 25. Thus the mean square of the x-momentumdefined in Figure 10, px =

√2m(E + ~ω)sinθcosϕ, becomes

(24) 〈p2x〉 =

2m∫ EFEF+φeff−~ω dE

∫ 1√EF+φeffE+~ω

d(cosθ)∫ 2π

0dϕ(E + ~ω)sin2θcos2ϕ∫

dE∫d(cosθ)

∫dϕ

Performing these integrals results a relatively simple relation for the photo-electric normalized divergence,

(25) ∆photo = βγσphotox′ =

√~ω − φeff

3mc2

The normalized divergence vs. photon energy for photo-electric emission is plottedin Figure 11 with 0, 50 and 100 MV/m applied fields. And the normalized cathodeemittance for photo-emission is simply,

(26) εphoto = σx

√~ω − φeff

3mc2

Plotting these expressions for the emittance and the quantum efficiency in Figure12 illustrates the strong correspondence between them as described above. A typicallaser wavelength for copper cathodes is 255 nm, where the QE given by Equation 25is 1.9× 10−4 and 0.30 microns/mm for the emittance. The QE in an operating RFphotocathode gun is nearly ten times lower( 3×10−5) and the measured emittanceis 0.6 microns/mm.

Thus far the discussion of photoelectric emission has assumed the free electrongas model which is appropriate for metals. However emission from semi-conductorcathodes is distinctly different, and their treatment is beyond the scope of thislecture. An in depth discussion of this and other emission phenomena can be foundin the recent book: Advances in Imaging and Electron Physics, Electron EmissionPhysics, Vol. 149, by K.L. Jensen.

6. Field Emission

Field emission occurs under the influence of very high fields of 109 V/m or more.Since the electrons quantum mechanically tunnel through the barrier, the highelectric fields are necessary to lower the barrier enough the achieve useful emissioncurrents.


Figure 11. The normalized divergence plotted as a function ofthe photon energy for applied fields of 0, 50 and 100 MV/m. The4.86 eV laser energy used for LCLS is shown by the vertical dash-line.

Field emission at electron temperatures above 1000 degK is referred to as thermal-field emission . This type of emission is used in thermionic rf guns which are com-monly used as injectors for 3rd generation storage ring light sources. The effectof temperature is greatest at low fields and high temperatures (+1000 degK andhigher) with the current density increasing by more than an order of magnitudeover that at ambient (∼300 degK) temperature.

The field emission current density is given by

(27) jfield =∫n(Ex, T )D(Ex, E0)dEx

where the supply function, n(Ex, T ), is the flux of electrons incident upon thebarrier with energies between Ex and Ex + dEx. The barrier as shown in Figure 2is determined by the work function, the image charge and the applied electric field,E0. The transmission of electrons through this barrier is given by the transparencyfunction, D(Ex, E0). This function can be solved using the WKB approximationfor a barrier produced by the image charge and the applied field,

(28) φSchottky(x) = − e2

16πε0x− eE0x


Figure 12. The photo-electric normalized divergence (top, red)and the quantum efficiency (bottom, red) vs. photon wavelengthfor copper. The green ellipses show the range of QE and cathodeemittance measured during commissioning of the LCLS injector.

The resulting transparency function is

(29) D(Ex, E0) = exp

[−8π√

2m3he

E3/2x

E0θ

(√e3E0

φwork

)]where θ(y) is the Nordheim function (not to be confused with the angle, θ). Al-though the θ -function is strongly dependent upon y and is mathematically compli-cated, it monotonically varies from 1 to 0 on the interval 0 < y < 1 and to a verygood approximation can be represented by

(30) θ(y) = 1− 0.142y − 0.855y2

The supply function for a Fermi-Dirac electron gas was also derived by Nordheim,

(31) n(Ex, T ) =4πmkBT

h3ln(

1 + eEx−EFkBT

)Combining the supply and transparency functions the electron energy spectrum

is simply,

(32) Nfield(Ex, E0, T ) = n(Ex, T )D(Ex, E0)


The combination of these three functions is plotted in Figure 13 vs. the electronenergy.

Figure 13. The spectrum of the emitted electrons results fromthe overlap of the barrier transparency (blue) with the supply func-tion (red). In this case, the supply function is for a Fermi-Diracdistribution at a temperature of 300 degK .

In field emission the electron yield is exponentially sensitive to the external fieldand any significant current requires fields in excess of 109 V/m as seen in Figure 14.Such high fields are achieved by using pulsed high voltages and/or field-enhancing,sharp emitters. Knowledge of the energy spectra allows numerical computation ofthe rms energy spread and the field emission emittance which are plotted in Figure15 for external fields between 109 and 1010 Volts/m.

7. Discussion of the Three Emission Processes

The last three sections derived the cathode emittances for the three emission pro-cesses: thermionic, photo-electric and field emission. Rather than using the term,thermal emittance, we prefer to using cathode emittance for the emission processes.And to specifically derive the cathode emittance for each of the three emission pro-cesses. The cathode emittance for thermionic emission is approximately 0.3 mi-crons/mm for a cathode temperature of 2500 degK. The photo-electric emittanceranges between 0.5 to 1 micron/mm depending upon the photon wavelength andwas shown to be proportional to the quantum efficiency. The field-emission emit-tance varies from 0.5 to 2 microns/mm for fields corresponding to fields of 109 to


Figure 14. Electron spectra for field emission electrons for vari-ous applied fields. Left: Electron emission spectra plotted with alinear vertical scale and with arbitrarily normalization to illustratethe spectral shapes. Right: The spectral yields plotted logarithmi-cally to illustrate the strong dependence of yield and shape uponapplied field.

Figure 15. The energy spread and field emission emittance froma planar emitter as functions of the external field.

1010 V/m, and hence has larger emittance for the same source size than the othertwo processes.

In many cases electron guns combine field emission with the other two processesto produce electrons. A good example is photo-electric emission in a high field gun,where field-emission lowers the barrier height via the Schottky effect and increasesthe quantum efficiency.

Class Exercises:

1) Derive the relation for the Schottky potential ; 2) Compute the Schottkypotential for 100 MV/m. Compare this with the work function for copper.


Figure 16

8. The Refraction of Electrons at the Cathode Surface

Similar to photons, electrons change their angle or refract as they transit thecathode surface. This is a result of the boundary condition requiring conservation ofthe transverse momentum across the cathode-vacuum interface. Using the momentaand angles defined in Figure 17, the boundary condition is expressed as

(33) pinx = poutx

(34) pinx = pintotalsinθin = pouttotalsinθout = poutx

The total energy of the electron inside the cathode after absorbing the photon isE + ~ω. Thus the total momentum inside is,

(35) pintotal =√

2m(E + ~ω)

and outside the total momentum is

(36) pintotal =√

2m(E + ~ω − EF − φeff )

Inserting these expressions into Equation 34 and solving for the ratio of sines givesa formula similar to Snell’s law,

(37)sinθoutsinθin

=

√E + ~ω

E + ~ω − EF − φeff


Figure 17. Definitions of the electron momenta and angles at theboundary between the cathode and vacuum.

An electron approaching the boundary needs to have sufficient momentum toovercome the barrier,

(38) pz =√

2m(E + ~ω)cosθin ≥√

2m(EF + φeff )

This leads to the maximum escape angle, θmaxin ,

(39) cosθmaxin =

√EF + φeffE + ~ω

and

(40) sinθmaxin =

√E + ~ω − EF − φeff

E + ~ωthese relations are shown graphically in Figure 18.

Figure 18. Graphical relation between electron, Fermi and workfunction energies.

On the vacuum side of the boundary the maximum electron angle is,

(41) sinθmaxout = sinθmaxin

√E + ~ω

E + ~ω − EF − φeff−→ θmaxout = π/2


and the electron behaves just like a photon and undergoes total internal reflectionfor angles greater than the the critical angle, θmaxin .

Thus it is reasonable to associate the square root of these energies with thefollowing indices of refraction for the electron inside and outside the cathode, re-spectively,

(42) nin =√E + ~ω

(43) nout =√E + ~ω − EF − φeff

This analogy suggests focusing effects for the escaping electrons, as discussed inthe next section.

9. Focusing of Electrons by the Cathode-Vacuum Boundary

The bending of light by a curved boundary between media with different indicesof refraction is illustrated in Figure 19 and expressed as (see for example, ”Optics,”by Miles V. Klein, John Wiley & Sons, Inc, 1970, pp. 67-69),

(44) x1 =[

(n0 − n1)D1

n1R+ 1]x0 +

[D0 +

n0D1

n1+

(n0 − n1)D0D1

n1R

]θ0

(45) θ1 =n0 − n1

n0Rx0 +

[n0

−n1+

(n0 − n1)D0

n1R

]θ0

In comparison with the paraxial equations for ray-optics, the focal strength ofthe refracting boundary is,

(46)1f

=n1 − n0

n1R

Figure 19. Quantities used in the relations for refraction at andindexed boundary


Inserting the previously derived relations for the electron indices of refractiongives,

(47)1f

=1R

[1−

√E + ~ω

E + ~ω − (EF + φeff )

]Clearly the square root term is always greater than unity, making the quantity in thebrackets negative. Therefore surfaces with positive curvature will give the electronsa diverging angle, while negative curvature surfaces give them a converging angle.

Since the electrons are all emitted from a thin layer an optical depth thick, thenD0 ≈ 0 and Equations 44 and 45 become,

(48) x1 =[

(n0 − n1)D1

n1R+ 1]x0 +

n0D1

n1θ0

(49) θ1 =n0 − n1

n0Rx0 −

n0

n1θ0

Class Exercise: 1) Compute the surface curvature necessary to produce acollimated beam. 2) Derive the emittance growth due to the sinusoidal surfacegiven as: z(x) = Ax sin(2πfracxλx). Show that, similar to the thermal emittance,it scales linearly with beam size.

10. The Debye Length and Laminar Flow

Once free of the cathode, the electrons experience both external and internalmutual forces. The external forces are those imposed by RF, magnetic and electricfields. These are forces which the designer of the injectors can use to create andcontrol the beam. The internal forces results from the mutual interactions betweenthe electrons, which can be classified into collisions and smoothly varying forces.The collisional forces occur between nearest neighbors and are random, statisticalfluctuations related to the beam temperature. On a larger scale lengths the elec-trons will collectively move to screen any non-uniform of the electron distribution,resulting in a slowly, varying spatial field. This screening of the single-particleforces is called Debye shielding [Reiser, p 184; Jackson, 1st Edition, pp 339] and iseffective for distances greater than the Debye length, λD, which is defined as theratio of the beam’s random, thermal velocity to the plasma frequency, ωp =

√e2nε0m

, where n is the electron density and m is the electron mass,

(50) λD =〈v2x〉ωp

Assuming M-B statistics, the Debye length in the beam rest frame is

(51) λD =

√ε0kBTbe2n

The procedure for the relativistic transformation of this relation to the lab frameis ambiguous as described by Reiser who simply uses T = Tb/γ to obtain thelaboratory frame Debye length in terms of laboratory quantities, n, γ and thebeam rest frame temperature, Tb,

(52) λD =

√ε0γkBTbe2n


In some cases, and especially when the electrons have just escaped from the cathode,the electron statistics are not necessarily given by the M-B thermal distribution.

On length scales where the Debye length is large compared to the distance be-tween the electrons and comparable to the beam size, the collective space chargeinteractions are diminished and independent particle dynamics dominate. As anexample consider the 135 MeV, 1 nC beam in the LCLS injector with a radius of100 microns and a bunch length of 6 ps and a thermal temperature kBTb of 0.2 eV,for which λD is 5.1 microns. The inter-particle distance is 0.2 microns. Since theDebye length determines the shortest plasma wavelength, in this case, beam plasmawaves shorter than 5 microns are not possible. The shorter wavelength oscillationsare washed out by the thermal motion of the electrons. When the space chargewave velocity is close to the thermal velocity and the waves are Landau damped.(Jackson, 1st Edition, pp340.)

Class Exercise: Beginning with Equation 50, write a relation for the Debyelength of a photo-emitted beam.

11. The Space Charge Limit and Cathode Emittance

In this section it is shown that space charge limited emission for a cathodeemitting a short electron bunch evolves toward the Child-Langmuir law as thebunch length increases and the region between the bunch and cathode becomesfilled with electrons. A general expression for the space charge limit is derivedfirst for a short bunch near the cathode surface and then for a continuous currentof electrons streaming from the cathode, yielding the Child-Langmuir law withouthaving to solve Poisson’s second-order differential equation. This result is used tocompute the emittance for photo-electric emission at the space charge limit.

Consider the electron bunch of length δ when it is a short distance d fromthe cathode surface as shown in Figure 1. If d is small compared to the beam’stransverse dimension, then Gauss’s law is trivially integrated over the cylinder’ssurface to solve for the space charge electric field between the cathode and a thinsheet of electrons.

(53)∮S

~E · ~ndS =1ε0

∫V

ρ(~r)d 3r

And obtain the space charge field of a sheet beam,

(54) Ez =σ

ε0

We reach the space charge limit (SCL) when the space charge field, Ez equalsthe applied, external field, Ea. That is,

(55) σSCL ≡ ε0Ea

The space charge potential energy of the short bunch is obtained by a simple inte-gration along z,

(56)dφdz

= Ez =σ

ε0


Figure 20. The cathode and electron bunch shortly after emis-sion. The electron bunch is typically 10’s of microns long when thelaser pulse ends and the last electron is emitted. d is the distancefrom the cathode to the tail of a bunch which is δ long.

Thus the surface charge density corresponding to the SCL in terms of the potentialenergy is given by,

(57) σSCL =ε0φad

Next we derive the Child-Langmuir law, and begin by expressing the beam cur-rent density in terms of the charge density and beam velocity, βc,

(58) J = ρβc =σ

δβc

We assume the electrons have negligible initial velocity at the cathode surface, and

thus their kinetic energy equals the applied potential, βc =√

2eφam . This gives the

SCL peak current density of the short bunch as,

(59) JbunchSCL = ε0

√2em

φ3/2a

δd

When the electrons fill the region between the cathode and the head of the bunch,δ = d,

(60) JCLSCL = ε0

√2em

φ3/2a

d2

which is the well-known Child-Langmuir law for current flowing in a diode region.The space charge limited emittance is found by combining the beam size for a the

short bunch at the SCL with the normalized divergence derived for photo-electric


emission. If we assume the beam is transversely unform with radius a then

(61) a =√Qbunchε0πEa

Which has the root-mean-square size of

(62) σx =a

2=√Qbunch4πε0Ea

The normalized cathode emittance is given by

(63) ε = βγσxσx′

And substituting the normalized divergence for photo-electric emission results inthe SCL photo-electric emittance,

(64) εSCLphoto =

√Qbunch(~ω − φeff )

4πε0mc2Ea

Here φeff is the effective cathode work function which includes the Schottky effectand ~ω is the laser photon energy. For a bunch charge of 1 nC, an applied fieldof 50 MV/m, a laser energy of 4.86 eV and a copper cathode (φeff = 4.5eV ), thespace charge limited thermal emittance is

(65) εSCLphoto = 0.34 microns

12. Charge Limited Emission in RF Guns

When a transverse gaussian beam reaches the short bunch space charge limit,emission saturates and the charge becomes constant and independent of the laserenergy. This constant charge is over an area defined by Eqn. 61,and results in thecharge vs the drive laser energy shown in Figures 21 and 22. The charge at low laserenergy depends linearly on the laser energy while at high laser energy the chargerolls over due to the space charge limit. A larger laser size reduces the space chargefield and gives more charge for the same laser energy. The linear region QE limitsthe emission, while space charge limited emission occurs at the high laser fluenceas illustrated experimentally in the figures.

Class Exercise: Derive an expression for the emitted charge as a function of thelaser energy for a transverse gaussian distribution. Write the relation for the tran-sition between QE limited and space charge limited emission. [see:J. Rosenzweiget al., ”Initial measurements of the UCLA rf photoinjector”, NIM A341(1994)379-385; J.L. Adamski et al., ”Results of commissioning the injector and constructionprogress of the Boeing one kilowatt free-electron laser”, SPIE Vol. 2988, p158-169; R. Akre et al., ”Commissioning the Linac Coherent Light Source Injector,”PRST-AB 11, 030703(2008).]

13. The Envelope Equation and Beam Perveance

The envelope equation without external focusing is,

(66) r′′m =ε2

r3m+K

rm


Figure 21. The charge vs. the drive laser energy for 433 MHzRF photocathode gun operating with a peak cathode field of 25MV/m. The two sets of data are shown with curves correspondingto measured x and y laser sizes. The straight line gives the cathodeQE. (Adamski et al., SPIE Vol. 2988, p158-169.)

where K is the generalized perveance (Lawson, p117) and ε is the geometric emit-tance,

(67) Kgeneral =I

I0

2β3γ3

=ω2pa

2

2β2c2.

Otherwise the perveance is defined as (Reiser, p.)

(68) K =I

4πε0V 3/2(

2em

)1/2 =I

d2JCLSCL

Where the denominator is identified as the Child-Langmiur current from Eqn. 60.Therefore the perveance is defined as the beam current divided by the space chargelimited current, d2JCLSCL.

When the beam is in radial equilibrium, r′′e = 0, the geometric emittance can berelated to the perveance,

(69)ε2

r3e=K

re

(70) ε = re√K


Figure 22. Measurements of the charge vs. the laser energy forthe LCLS-gun operating at a peak cathode field of 115 MV/m.The data were taken with the same beam size (1.2 mm diameter)but for different QE.

(71) ε = re

√√√√ I

V 3/2

[1

4πε0(

2em

)1/2]

= re

√I

d2JCLSCL= re√K

Thus the square root of the perveance is the space charge limited geometric diver-gence.

introduction - uspasuspas.fnal.gov/materials/08ucsc/lecture2_emissionstatisticscathode...lecture 2:...

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